Question
A diagonal matrix is a square matrix with all zero entries above and below its main diagonal. Evaluate the determinant of each diagonal matrix. Make a conjecture based on your results.a. $\left[\begin{array}{ll}7 & 0 \\ 0 & 4\end{array}\right]$b. $\left[\begin{array}{rrr}-1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2\end{array}\right]$c. $\left[\begin{array}{cccc}2 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 3\end{array}\right]$
Step 1
For the matrix in part a, we have $a = 7$, $b = 0$, $c = 0$, and $d = 4$. So, the determinant is $7*4 - 0*0 = 28$. Show more…
Show all steps
Your feedback will help us improve your experience
Kyle Christian and 88 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero. Find the determinant of each diagonal matrix. Make a conjecture based on your results. (a) $\left[\begin{array}{ll}7 & 0 \\ 0 & 4\end{array}\right]$ (b) $\left[\begin{array}{rrr}-1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2\end{array}\right]$ (c) $\left[\begin{array}{rrrr}2 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 3\end{array}\right]$
Matrices and Determinants
The Determinant of a Square Matrix
A triangular matrix is a square matrix with all zero entries either above or below its main diagonal. Such a matrix is upper triangular when it has all zeros below the main diagonal and lower triangular when it has all zeros above the main diagonal. A diagonal matrix is both upper and lower triangular. Evaluate the determinant of each triangular matrix. Make a conjecture based on your results. (a) $\left[\begin{array}{rr}3 & -2 \\ 0 & 5\end{array}\right]$ (b) $\left[\begin{array}{rrr}3 & -7 & 1 \\ 0 & -5 & -9 \\ 0 & 0 & 5\end{array}\right]$ (c) $\left[\begin{array}{rrrr}4 & 0 & 0 & 0 \\ 3 & -3 & 0 & 0 \\ 3 & 6 & 5 & 0 \\ 2 & -2 & 1 & 2\end{array}\right]$
Linear Systems and Matrices
A triangular matrix is a square matrix with all zero entries either above or below its main diagonal. Such a matrix is upper triangular when it has all zeros below the main diagonal and lower triangular when it has all zeros above the main diagonal. A diagonal matrix is both upper and lower triangular. Evaluate the determinant of each triangular matrix. Make a conjecture based on your results. a. $\left[\begin{array}{rr}3 & -2 \\ 0 & 5\end{array}\right]$ b. $\left[\begin{array}{ccc}3 & -7 & 1 \\ 0 & -5 & -9 \\ 0 & 0 & 5\end{array}\right]$ c. $\left[\begin{array}{rrrr}4 & 0 & 0 & 0 \\ 3 & -3 & 0 & 0 \\ 3 & 6 & 5 & 0 \\ 2 & -2 & 1 & 2\end{array}\right]$
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD
